PROBABILITY CURVES 



103 



The series expansion (1) is determined by equating the integrands 

 of (2) and (3), 



-^ T a c ~ x e ~ a da = — -j= e ~ iP dt, 



r(c) V2t 



(4) 



and solving for positive values of a with the condition that / = — °° 

 when a = 0. 



Let c = 



1_ 



b 2 ' 



a =L Qi Q = eL) 



r(c) 



V27 



= i(6 2 g)- 1 /6 2 e A/ ) 



R=- — ro \t 2 + M. 



(5) 



Substituting these values (5) in equation (4), 

 L' = be R , 



where L' is written for dL/dt. 



(6) 



30 QO CO 00 



Let L = Vl s i ! , M= ]T Msb*, R= V^& , Q=^QJ>*, (7) 



where the coefficients are polynomials in / (constants in the case of 

 the series for M). Upon substituting these series expansions for the 

 functions in the last equality of (5) and equating coefficients of like 

 powers of b, we obtain 



=()o-£o-l, 



=Qi-L lt 



Ro = Q2-L 2 -ht 2 + M 0< 



R 1 =Q S -L 3 +M l , 



R 2 ^Q i -L i -hM 2 , 



R„ = Q n+ 2-L n+2 + M n , (w = l, 2, 3 . . . ). 



(8) 



From (5) we obtain Q = e Lo , and then L -\-\ = e Lo , the only real 

 solution of which is L o = 0, and therefore, Q —l- 



